Evolution of the cosmic matter density field with a primordial magnetic field

A cosmological magnetic field affects the time evolution of the cosmic matter density field. The squared Alfven velocity of the cosmic fluid is proportional to an ensemble average energy density of a primordial magnetic field (PMF), and it prevents the matter density field from collapsing in the horizon scale. The matter-radiation equality time also is delayed by the presence of the ensemble average energy density of the PMF. The ensemble average energy density of the PMF also affects the matter power spectrum (MPS) through the Meszaros effect and the potential decay. Since the ensemble average energy density of the PMF is not a first order perturbation but a zero order source in the linear perturbation equations for the cosmology, to correctly understand the overall effect of the PMF on the MPS, we should significantly revise previous approaches to research for the MPS with the PMF by considering both the effects of the zero and first order sources from the PMF in the linear perturbation theory. We apply the effects of the zero order sources from the PMF to theoretical computations of the MPS for the first time. We also analyze the overall PMF effect on the MPS. The CMB polarization is affected by the weak lensing. The weak lensing is determined by the MPS. Therefore, we have to consider the zero order sources of the PMF to gain a correct understanding not only of the MPS but also the CMB polarization.

Figure 1

The effects of the PMF on the MPS in the case of the fixed PMF strength as $B_{\lambda }=10\mathrm{nG}$. The thin dotted curve in this figure is the theoretical result from the Planck and WMAP 9 yr best-fit parameter in the $\mathrm{\Lambda }\mathrm{CDM}$ model [89, 90]. The dots with error bars show 2dF [97]. The solid, dotted, dashed, and dash-dotted curves are the theoretical result with the PMF effects of $(n_{\mathrm{B}},{\rho }_{\mathrm{MF}}/{\rho }_{\gamma })=(-2.99,1.00\times {10}^{-5})$, $(-2.00,5.84\times {10}^{-4})$, $(-1.00,9.65\times {10}^{-3})$, and $(0.00,8.63\times {10}^{-2})$, respectively.

Figure 2

The effects of the zero and first order sources of the PMF on the MPS in the case of the fixed PMF strength as $B_{\lambda }=10\mathrm{nG}$. The thin dotted curves in this figure are the theoretical result from the Planck and WMAP 9 yr best-fit parameter in the $\mathrm{\Lambda }\mathrm{CDM}$ model [89, 90]. The dashed curves show the MPS with the first order sources and without the zero order sources from PMF. The solid curves show the MPS with the first and the zero order sources from PMF. The PMF parameters sets in the panels (a), (b), (c), and (d) are $(n_{\mathrm{B}},{\rho }_{\mathrm{MF}}/{\rho }_{\gamma })=(-2.99,1.00\times {10}^{-5})$, $(-2.00,5.84\times {10}^{-4})$, $(-1.00,9.65\times {10}^{-3})$, and $(0.00,8.63\times {10}^{-2})$, respectively.

Figure 3

The effects of the PMF on the MPS in the case of ${\rho }_{\mathrm{MF}}/{\rho }_{\gamma }=5.0\times {10}^{-4}$. The thin dotted curves in this figure are the theoretical result from the Planck and WMAP 9 yr best-fit parameter in the $\mathrm{\Lambda }\mathrm{CDM}$ model [89, 90]. The solid curves in panels (a), (b), and (c) show the MPS with the zero and first order sources from the PMF. The dash-dotted curves in panels (a), (b), and (c) show the MPS with first order sources from the PMF and without zero order sources from the PMF. The PMF parameters in panels (a), (b), and (c) are $(n_{\mathrm{B}},B_{\gamma })=(-2.99,70.9\mathrm{nG})$, $(-2.50,25.0\mathrm{nG})$, and $(-2.00,9.11\mathrm{nG})$, respectively.